\section{Discussion of results}

The heat transfer production term in the entropy equation includes the square of the temperature gradient. By looking at the fast chemistry formulation, eq. \ref{eq:FastTmixfrac}, and its derivative Eq. \ref{eq:FastderivT} we see that increasing the oxidizer temperature, $T_2$, decreases the temperature gradient in the reacting flow. Further inspection of the entropy production from heat transfer, eq. \ref{eq:Qentropy}, yields that if $T_1$ is held constant this term varies with $\Delta (T)^2$ where $\Delta(T)=T_1-T_2$

%$T_2^{-2}*\rho_2^{-2}*rho^2*derivT^{2} \propto T_2^-2*T_2^2*(\Delta T+Q)^2 \propto (\delta T)^2$

Therefore preheating air should decrease the irreversibilities from internal heat transfer significantly. The entropy production from mass transfer and chemical reaction will be reduced slightly since the average density decreases and thus the $\overline{\rho \chi}$ term is lowered. 

Lowering the concentration gradients will lead to a reduction of the contribution of entropy production from mass transfer.

When using curvefits the rate of entropy production for entropy production through all the three evaluated paths had higher values. In \ref{tab:curvefitTotalEntropy} we see that the entropy production from chemical reaction is drastically higher than for heat transfer and mass transfer.

$OH$ only exists in the reaction zone as an intermediate species of reaction. The formation of $OH$ follows the steps shown below.

\begin{eqnarray}
\text{Step 1: } & H_2 + O_2 \rightarrow HO_2+H\\
\text{Step 2: } & H+ O_2 \rightarrow OH + O\\
\text{Step 3: }	& O+H_2 \rightarrow OH+ H
\label{eq:OH-form}
\end{eqnarray} 


The distribution of $OH$ in the flame can therefore give an indication of the position and thickness of the reaction zone. From Fig. \ref{fig:SandiaH2YOH} it can be seen that the concentration of $OH$-radicals is highest in a thin shell at the lower part of the flame. The entropy production from chemical reaction when using the fast chemistry assumption, Fig. \ref{fig:fastchem}, and for the curve fits ,Fig. \ref{fig:curvefitchem}, have maximum values in roughly the same region. 

From \cite{MS} the specific chemical exergy, $a^{ch}$ of $H_2$ is found from the reaction $H_2+\frac{1}{2} O_2 \rightarrow H_2O$ as 

\begin{equation}
a^{ch}_{H_2}=[g_{H_2}+\frac{1}{2}g_{O_2}-g_{H_2O}](T_{ref},p_{ref})+\mathcal{R} T_{ref} ln \left[\frac{(X^e_{O_2})^{\frac{1}{2}}}{(X^e_{H_2O})} \right]
\label{eq:chemexergyH2}
\end{equation}

where the superscript $e$ means that the mole fractions should be considered in the surrounding environment. $g$ is the specific Gibbs function given as $g=h-Ts$. The mass flow rate of the Sandia $H_2$-flame is $\dot{m}_{H_2}=\unit{0.000264}{\kilogram \per \second}$. 

The chemical exergy of hydrogen evaluated at $T_{ref}=298K$, $p_{ref}=$1 atm with $(X^e_{H_2O})=0.0115$ (given in the experiment documentation \cite{SandiaH2He}), $X^e_{O_2}=0.208$ and with values for the Gibbs function of formation from Tab A-25 in \cite{MS} is $[-(-228590)+8.314 \cdot 298 ln(0.208^{0.5}/0.0115)]/M_{H_2}=\unit{117910}{\kilo \joule \per \kilogram}$. This is 98.3\% of the lower heating value for hydrogen. The exergy of the incoming fuel flow, $\dot{A}_{in}$ in the equation for second law efficiency Eq. \ref{eq:Second law efficiency}, is then found as the mass flow rate multiplied with the specific chemical exergy of the fuel (the contribution of kinetic energy of the flow is relatively small). The second law efficiency is found to be 76.0 \%
for curvefit approximation and 98.7 \% when fast chemistry is assumed. In the current literature the exergy destruction in the combustor is believed to be the range 20-30\% so the results from the simulations with curvefitted experimental data are plausible, but they should indeed be verified.

\cite{Nishida2002} found that for laminar diffusion flames, heat conduction is the largest contributor to the total exergy destruction.  

The larger contribution of chemical reaction to the total entropy production can be somewhat caused

Also the model used for $\overline{\rho}\tilde{\chi}$  Eq.\ref{eq:Scalar dissipation} has not been proved mathematically, but it is believed that it can approximate the scalar dissipation rate to reasonable accuracy, as there are other large sources of inaccuracy with this modelling approach.

The discretization method for volume integration of the the entropy production in a 2D plane could magnify errors.

Inaccuracies in the measurements could also be a source of error.